Finite energy global well-posedness of the Chern-Simons-Higgs equations in the Coulomb gauge

In a recent paper, Selberg-Tesfahun proved that the abelian Chern-Simons-Higgs system (CSH) is globally well-posed for finite energy initial data under the Lorenz gauge condition. It has been suspected by Huh, however, that such a result should hold in the Coulomb gauge as well. In this note, we give an affirmative answer to this question by first establishing low regularity local well-posededness of (CSH) in the Coulomb gauge for initial data set $(f,g)\in H_x^{\gamma }\times H_x^{\gamma -1}$ for any $\gamma > 3/4$. Then by conservation of energy, global well-posedness for (CSH) in the energy space $(f,g)\in H_x^1\times L_x^2$ follows rather immediately.